The constant can be characterized in many different ways. For example, e can be defined as the unique positive number a such that the graph of the function y = ax has unit slope at x = 0.[3] The function f(x) = ex is called the (natural) exponential function, and is the unique exponential function equal to its own derivative. The natural logarithm, or logarithm to basee, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one (see image). There are alternative characterizations.

Sometimes called Euler's number after the SwissmathematicianLeonhard Euler, e is not to be confused with γ, the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of the symbol e is said to have been retained in his honor.[4] The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.[5]

The number e is of eminent importance in mathematics,[6] alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995... (sequence A001113 in the OEIS).

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[5] However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli in 1683,[7][8] who attempted to find the value of the following expression (which is in fact e):

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach of 25 November 1731.[9][10] Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,[11] and the first appearance of e in a publication was in Euler's Mechanica (1736).[12] While in the subsequent years some researchers used the letter c, the letter e was more common and eventually became standard.[citation needed]

The constant has been historically typeset as "e", in italics, although the ISO 80000-2:2009 standard recommends typesetting constants in an upright style.

An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.254 = $2.4414..., and compounding monthly yields $1.00 × (1 + 1/12)12 = $2.613035... If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00×(1 + 1/n)n.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $2.692597..., while compounding daily (n = 365) yields $2.714567..., just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach $2.7182818...

More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. (Here R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05.)

Graphs of probability P of not observing independent events each of probability 1/n after n Bernoulli trials, and 1 - P vs n ; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 1/e.

The number e itself also has applications to probability theory, where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in n and plays it n times. Then, for large n (such as a million) the probability that the gambler will lose every bet is approximately 1/e. For n = 20 it is already approximately 1/2.79.

This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in one million chance of winning. Playing one million times is modelled by the binomial distribution, which is closely related to the binomial theorem. The probability of winning k times out of a million trials is:

Another application of e, also discovered in part by Jacob Bernoulli along with Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem:[13]n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into n boxes, each labelled with the name of one guest. But the butler does not know the identities of the guests, and so he puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. The answer is:

As the number n of guests tends to infinity, pn approaches 1/e. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is n!/e rounded to the nearest integer, for every positive n.[14]

The stated result follows because the maximum value of x−1ln⁡x{\displaystyle x^{-1}\ln x} occurs at x=e{\displaystyle x=e} (Steiner's problem, discussed below). The quantity x−1ln⁡x{\displaystyle x^{-1}\ln x} is a measure of information gleaned from an event occurring with probability 1/x{\displaystyle 1/x}, so that essentially the same optimal division appears in optimal planning problems like the secretary problem.

The constraint of unit variance (and thus also unit standard deviation) results in the 1/2 in the exponent, and the constraint of unit total area under the curve ϕ(x) results in the factor 1/2π{\displaystyle \textstyle 1/{\sqrt {2\pi }}}.[proof] This function is symmetric around x = 0, where it attains its maximum value 1/2π{\displaystyle \textstyle 1/{\sqrt {2\pi }}}, and has inflection points at x = ±1.

The functions x↦ax{\displaystyle x\mapsto a^{x}} are shown for a=2{\displaystyle a=2} (dotted), a=e{\displaystyle a=e} (blue), and a=4{\displaystyle a=4} (dashed). They all pass through the point (0,1),{\displaystyle (0,1),} but only ex{\displaystyle e^{x}} is there tangent to the red line of slope 1.{\displaystyle 1.}

The parenthesized limit on the right is independent of the variable x: it depends only on the base a. When the base is set to e, this limit is equal to 1, and so e is symbolically defined by the equation:

ddxex=ex.{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.

Another motivation comes from considering the derivative of the base-alogarithm,[17] i.e., of logax for x > 0:

The logarithm with this special base is called the natural logarithm and is denoted as ln; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

There are thus two ways in which to select such special numbers a. One way is to set the derivative of the exponential function ax equal to ax, and solve for a. The other way is to set the derivative of the base a logarithm to 1/x and solve for a. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for a are actually the same, the number e.

for all real x, with equality if and only if x = 0. Furthermore, e is the unique base of the exponential for which the inequality ax ≥ x + 1 holds for all x.[19] This is a limiting case of Bernoulli's inequality.

It is conjectured that e is normal, meaning that when e is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).

Because this series keeps many important properties for ex even when x is complex, it is commonly used to extend the definition of ex to the complex numbers. This, with the Taylor series for sin and cos x, allows one to derive Euler's formula:

In addition to exact analytical expressions for representation of e, there are stochastic techniques for estimating e. One such approach begins with an infinite sequence of independent random variables X1, X2..., drawn from the uniform distribution on [0, 1]. Let V be the least number n such that the sum of the first n observations exceeds 1:

In contemporary internet culture, individuals and organizations frequently pay homage to the number e.

For instance, in the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is e billion dollars rounded to the nearest dollar. Google was also responsible for a billboard[36] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read "{first 10-digit prime found in consecutive digits of e}.com". Solving this problem and visiting the advertised (now defunct) web site led to an even more difficult problem to solve, which in turn led to Google Labs where the visitor was invited to submit a résumé.[37] The first 10-digit prime in e is 7427466391, which starts at the 99th digit.[38]

^Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston. ISBN0-03-029558-0.

^ abJacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for e. See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the Journal des Savants (Ephemerides Eruditorum Gallicanæ), in the year (anno) 1685.**), Acta eruditorum, pp. 219–223. On page 222, Bernoulli poses the question: "Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?" (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would he be owed [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si a=b, debebitur plu quam 2½a & minus quam 3a." ( … which our series [a geometric series] is larger [than]. … if a=b, [the lender] will be owed more than 2½a and less than 3a.) If a=b, the geometric series reduces to the series for a × e, so 2.5 < e < 3. (** The reference is to a problem which Jacob Bernoulli posed and which appears in the Journal des Sçavans of 1685 at the bottom of page 314.)